What is the integral of 1/(a^2e^x+e^{-x)} ?

The integral of 1/(a^2e^x+e^{-x)} is -1/a arctan((e^{-x})/a)+C

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integral of 1/(a^2e^x+e^{-x)}

integral

\frac{1}{a ^{2} e ^{x} + e ^{- x}}

- \frac{1}{a} arctan (\frac{e ^{- x}}{a}) + C

Solution steps

\int \frac{1}{a ^{2} e ^{x} + e ^{- x}} d x

Apply u-substitution

: \int - \frac{1}{a ^{2} + u ^{2}} d u

= \int - \frac{1}{a ^{2} + u ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{1}{a ^{2} + u ^{2}} d u

Apply Integral Substitution

= - \int \frac{1}{a ( v ^{2} + 1 )} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{a} \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= - \frac{1}{a} arctan (v)

Substitute back

= - \frac{1}{a} arctan (\frac{e ^{- x}}{a})

Add a constant to the solution

= - \frac{1}{a} arctan (\frac{e ^{- x}}{a}) + C

y + 1

\frac{( 3 x - 6 )}{( 3 x + 1 ) ( x - 1 )}

\frac{x}{x ^{2} + y ^{2}}

e^{4} x - 3

1^{22} x \cdot x^{2} - 11

What is the integral of 1/(a^2e^x+e^{-x)} ?
The integral of 1/(a^2e^x+e^{-x)} is -1/a arctan((e^{-x})/a)+C